Integrability Breaking and Coherent Dynamics in Hermitian and Non-Hermitian Spin Chains with Long-Range Coupling

Unraveling the mechanisms of ergodicity breaking in complex quantum systems is a central pursuit in nonequilibrium physics. In this work, we investigate a one-dimensional spin model featuring a tunable long-range hopping term, $H_{n}$, which introduces nonlocal interactions and bridges the gap between Hermitian and non-Hermitian regimes. Through a systematic analysis of level-spacing statistics, Krylov complexity, and entanglement entropy, we demonstrate that $H_{n}$ acts as a universal control parameter driving the transition from integrability to quantum chaos. Specifically, increasing the strength of $H_{n}$ induces a crossover from Poissonian to Gaussian Orthogonal Ensemble statistics in the Hermitian limit, and similarly triggers chaotic dynamics in the non-Hermitian case. Most remarkably, despite the onset of global chaos, we identify a tower of exact nonthermal eigenstates that evade thermalization. These states survive as robust quantum many-body scars, retaining low entanglement and coherent dynamics even under strong non-Hermitian perturbations. Our findings reveal a universal mechanism by which long-range and non-Hermitian effects reshape quantum ergodicity, offering new pathways for preserving quantum coherence in complex many-body systems.

💡 Research Summary

This paper presents a comprehensive study on the interplay between integrability, quantum chaos, and coherence in a one-dimensional spin-1 chain featuring a tunable long-range hopping term. The model Hamiltonian incorporates standard Heisenberg exchange (H_h), a chiral three-spin interaction (H_c), a Zeeman term (H_z), and a crucial nonlocal term H_n. The coupling strength J_n for H_n can be complex, serving as a unified control parameter that bridges the Hermitian and non-Hermitian regimes.

The authors first establish that the model hosts an exact, analytically solvable “tower” of ferromagnetic states. These states are fully symmetric (zero crystal momentum) and are generated by repeatedly applying the total spin-lowering operator to the fully polarized state. Remarkably, these states are exact eigenstates of the full Hamiltonian for any strength of H_n, as the antisymmetric structure of H_n yields a zero expectation value for this symmetric tower. Their energies depend linearly on magnetization.

In the Hermitian limit (real J_n), the system’s integrability is probed using the average level-spacing ratio ⟨r⟩. As J_n increases, ⟨r⟩ transitions from the Poisson value (~0.386, integrable) to the Gaussian Orthogonal Ensemble (GOE) value (~0.536, chaotic), signaling a breakdown of integrability driven by H_n. Intriguingly, at very large J_n, the system shows signs of re-entering a more regular regime. A phase diagram in the (J_c, J_n) plane reveals that chaos emerges only when both non-integrable couplings are sufficiently strong.

Despite the onset of global quantum chaos in the spectrum, the exact tower of states persists as quantum many-body scars (QMBS). These scar states exhibit low bipartite entanglement entropy (scaling logarithmically with system size), violating the Eigenstate Thermalization Hypothesis (ETH). Dynamically, when the system is initialized in a state with high overlap on this scar subspace, it exhibits long-lived coherent revivals in fidelity, in stark contrast to the rapid thermalization expected from the chaotic background.

The analysis is then extended to the non-Hermitian regime (complex J_n). Since real-level statistics are inapplicable, the authors employ the complex spacing ratio (CSR) and find an analogous chaos transition in the complex plane. Crucially, the scar tower remains robust even under strong non-Hermitian perturbations, retaining its low-entanglement character and coherent dynamical signatures.

The central finding is the dual role of the long-range term H_n: it acts as a universal integrability-breaking driver, yet it simultaneously preserves a protected subspace of exact nonthermal eigenstates due to a specific symmetry. This reveals a universal mechanism where long-range and non-Hermitian effects can reshape quantum ergodicity without destroying all quantum coherence. The work suggests new pathways for engineering and preserving coherent dynamics in complex, open quantum many-body systems, with potential implications for quantum simulation and information technologies.